Curing the unphysical behaviour of NLO quarkonium production at the LHC and its relevance to constrain the gluon PDF at low scales

We address the unphysical energy dependence of quarkonium-hadroproduction cross sections at Next-to-Leading Order (NLO) in α s which we attribute to an over-subtraction in the factorisation of the collinear singularities inside the PDFs in the MS ¯ scheme. Such over- or under-subtractions have a limited phenomenological relevance in most of the scattering processes in particle physics. On the contrary, it is particularly harmful for P T -integrated charmonium hadroproduction which renders a wide class of NLO results essentially unusable. Indeed, in such processes, α s is not so small, the PDFs are not evolved much and can be rather flat for the corresponding momentum fractions and, finally, some process-dependent NLO pieces are either too small or too large. We propose a scale-fixing criterion which avoids such an over-subtraction. We demonstrate its efficiency for η c , b but also for a fictitious light elementary scalar boson. Having provided stable NLO predictions for η c , b P T -integrated cross sections, σ η Q NLO , and discussed the options to study η b hadroproduction, we argue that their measurement at the LHC can help better determine the gluon PDF at low scales and tell whether the local minimum in conventional NLO gluon PDFs around x = 0.001 at scales below 2 GeV is physical or not.